Linear Operators: General theory |
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Page 424
... closed . Hence tK = ^ x , vex A ( x , y ) Ca € , xex B ( x , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space * of the B - space X is compact in the X topology of X * . PROOF . By ...
... closed . Hence tK = ^ x , vex A ( x , y ) Ca € , xex B ( x , x ) is also closed . Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space * of the B - space X is compact in the X topology of X * . PROOF . By ...
Page 429
... closed unit sphere of X is X - closed . PROOF . This follows from the preceding theorem and Corollary 2.14 . Q.E.D. 8 COROLLARY . If X is a B - space , a linear subspace YC X * is X - closed if and only if there exists an X - closed ...
... closed unit sphere of X is X - closed . PROOF . This follows from the preceding theorem and Corollary 2.14 . Q.E.D. 8 COROLLARY . If X is a B - space , a linear subspace YC X * is X - closed if and only if there exists an X - closed ...
Page 488
... closed range , then UX *** Y. PROOF . Let 0 y € Y and define T = Γ { y * \ y * € Y * , y * y = 0 } . Then I is -closed in Y * . - = Suppose , for the moment , that U * T is X - closed and different from U ** . From Corollary V.3.12 it ...
... closed range , then UX *** Y. PROOF . Let 0 y € Y and define T = Γ { y * \ y * € Y * , y * y = 0 } . Then I is -closed in Y * . - = Suppose , for the moment , that U * T is X - closed and different from U ** . From Corollary V.3.12 it ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ